The Ultimate Newcomb’s Problem

You see two boxes and you can ei­ther take both boxes, or take only box B. Box A is trans­par­ent and con­tains $1000. Box B con­tains a visi­ble num­ber, say 1033. The Bank of Omega, which op­er­ates by very clear and trans­par­ent mechanisms, will pay you $1M if this num­ber is prime, and $0 if it is com­pos­ite. Omega is known to se­lect prime num­bers for Box B when­ever Omega pre­dicts that you will take only Box B; and con­versely se­lect com­pos­ite num­bers if Omega pre­dicts that you will take both boxes. Omega has pre­vi­ously pre­dicted cor­rectly in 99.9% of cases.

Separately, the Numer­i­cal Lot­tery has ran­domly se­lected 1033 and is dis­play­ing this num­ber on a screen nearby. The Lot­tery Bank, like­wise op­er­at­ing by a clear known mechanism, will pay you $2 mil­lion if it has se­lected a com­pos­ite num­ber, and oth­er­wise pay you $0. (This event will take place re­gard­less of whether you take only B or both boxes, and both the Bank of Omega and the Lot­tery Bank will carry out their pay­ment pro­cesses—you don’t have to choose one game or the other.)

You pre­vi­ously played the game with Omega and the Numer­i­cal Lot­tery a few thou­sand times be­fore you ran across this case where Omega’s num­ber and the Lot­tery num­ber were the same, so this event is not sus­pi­cious.

Omega also knew the Lot­tery num­ber be­fore you saw it, and while mak­ing its pre­dic­tion, and Omega like­wise pre­dicts cor­rectly in 99.9% of the cases where the Lot­tery num­ber hap­pens to match Omega’s num­ber. (Omega’s num­ber is cho­sen in­de­pen­dently of the lot­tery num­ber, how­ever.)

You have two min­utes to make a de­ci­sion, you don’t have a calcu­la­tor, and if you try to fac­tor the num­ber you will be run over by the trol­ley from the Ul­ti­mate Trol­ley Prob­lem.

Do you take only box B, or both boxes?